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A contentious injury-time equaliser from James Tarkowski earned the Blues a 2-2 draw in the Reds' final visit to Goodison almost 131 years after their first clash at the stadium.
Oliver had irked both sets of supporters with a series of debatable calls throughout the game, but particularly upset Liverpool over allowing the late leveller - a decision upheld after a VAR check - despite what they claimed was a push on Ibrahima Konate by Beto during the build-up.
And after a furore sparked by Everton midfielder Abdoulaye Doucoure goading the away fans after the final whistle, the referee sent off Doucoure along with Curtis Jones, Liverpool boss Arne Slot and his assistant Sipke Hulshoff.
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Everton's equaliser came during an extension of the initial additional time after play was stopped with Liverpool on the attack following a collision between Blues duo Charly Alcaraz and Jarrad Branthwaite.
And Bradley said: "I don't think they should have got the ball back afterwards but that gave them one more chance.
The fans are right on top of you and they are giving you a bit of stick and it's a difficult game to play in, but I thought we did well and dealt with it well."
Bradley was replaced by Trent Alexander-Arnold in the second half shortly after the Everton crowd were riled by a foul challenge by the right-back, who had earlier been booked for a tackle on Vitalii Mykolenko.
"I felt like I had to be careful, especially after getting the yellow card," he said.
"I don't think I deserved the yellow card but I don't think we should talk about the referee as I'll probably get myself into trouble.
It was disappointing to get the yellow card and I just had to be good from then on in and make sure I didn't get sent off."
Goals in either half from Alexis Mac Allister and Mohamed Salah had seen Liverpool wipe out Beto's early opener.
He is a fantastic player and we are so lucky to have him."
Speaking to LFC TV, Bradley added: "It was bitterly disappointing at the end to lose the three points but we just have to bounce back on Sunday (at home to Wolves) and get another three points there.
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"It's a good position but we just have to keep going, keep working hard every day and keep improving.
We have to keep our heads down and keep going.
"We know it will be a difficult game against Wolves as they are a good team.
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